What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, 

> What properties are common to  Fourier transforms of all characteristic functions?


Here are a few trivial properties; what other properties are known?

 - $\widehat{1_A}$ is a bounded continuous function converging to zero.
 - $\widehat{1_A}$ is in $L^2$.

> What interesting functional analytic properties  does the set
$\{\widehat{1_A}: A \in \mathbb{R}^n\}$ of Fourier transforms of characteristic functions have?

At least it is closed in $L^2$ (just apply Plancherel's formula and use the fact that an $L^2$-limit of characteristic functions is a characteristic function). Is it closed in other norms? Is it dense in some interesting spaces (if we are allowed to multiply the functions by a constants)? For which $p$ is the Fourier transform a bounded operator from our set to $L^p$?

> How does the regularity (in a vague sense) of $A$ affect on the regularity of $\widehat{1_A}$? 

Here are a few easy remarks:

 - If $A$ is a finite union of intervals, then $\hat{1}_A(\xi)$ is a trigonomteric polynomial divided by $\xi$, so it is in every $L^p,p>1$ but not absolutely integrable.
 - If $A$ is bounded, the Fourier transform is an entire function in $\mathbb{C}^n$.